Enter Transformation Values
Formula Used
Main model: y = a · f(b(x - h)) + k
Parent input: u = b(x - h)
Final output: y = a · f(u) + k
The value a changes output height. It can stretch, compress, or reflect the graph. The value b changes input width. It can compress, stretch, or reflect the graph horizontally. The value h moves the graph left or right. The value k moves it up or down.
How to Use This Calculator
Select a parent function first. Then enter values for a, b, h, and k. Use the graph range fields to control the visible x values. Add an x value in the evaluation field when you need one exact answer. Press the calculate button. The result panel appears above the form. Use CSV for spreadsheet work. Use PDF for a clean report.
Example Data Table
| Parent Function | a | b | h | k | x | u | Transformed y |
|---|---|---|---|---|---|---|---|
| x² | 2 | 0.5 | 3 | -4 | 7 | 2 | 4 |
| x² | 2 | 0.5 | 3 | -4 | 3 | 0 | -4 |
| x² | 2 | 0.5 | 3 | -4 | -1 | -2 | 4 |
Understanding Function Transformations
A function transformation changes a parent graph without rebuilding it from zero. The parent function keeps its basic shape. The transformed function moves, flips, expands, or compresses that shape.
This calculator uses the common model y = a f(b(x - h)) + k. The value h moves the graph right or left. The value k moves it up or down. The value a controls vertical stretch, vertical compression, and reflection over the x-axis. The value b controls horizontal scaling and reflection over the y-axis.
Separate Inside and Outside Changes
Transformations are easier when you separate outside and inside changes. Outside changes affect output values. These include a and k. Inside changes affect input values. These include b and h. Because inside changes work before the parent function, their visual effect can feel reversed. A larger b often creates horizontal compression, not expansion.
Use Tables and Graphs Together
The table is useful for checking exact values. Each row shows an x value, the internal parent input, the parent output, and the transformed output. This helps students see how one point moves from the original graph to the new graph.
The graph gives a fast visual check. The parent curve and transformed curve can be compared across the same x range. Undefined values are skipped, so square root, logarithm, and reciprocal functions stay clear.
Build Better Algebra Checks
Use this tool for algebra homework, lesson planning, graph checks, or quick exploration. Try simple changes first. Set a to one, b to one, h to zero, and k to zero. Then change one value at a time. Notice how the graph responds.
For domain work, watch the internal input column. A parent function may have limits. Square root needs a nonnegative input. Logarithm needs a positive input. Reciprocal cannot use zero. The transformed domain changes because b(x - h) changes what input reaches the parent function.
The downloadable table makes results easy to save. Use the CSV file for spreadsheets. Use the PDF report for notes or classroom records. The evaluation field is helpful for one target point. Enter the x value you need. The calculator reports the parent input, parent output, final value, and transformation notes. This supports both numeric answers and graph explanations with fewer mistakes.
FAQs
1. What is a function transformation?
It is a change made to a parent function. The graph may move, flip, stretch, or compress while keeping the parent function’s main shape.
2. What does a control?
The value a controls vertical scaling. If a is negative, the graph reflects over the x-axis. Larger absolute values create vertical stretching.
3. What does b control?
The value b controls horizontal scaling inside the function. A negative b reflects the graph over the y-axis. Large values usually compress width.
4. Why does h shift the graph right?
In the model f(x - h), a positive h makes the parent input smaller. The graph must move right to produce the same parent input.
5. Why are some table values undefined?
Some parent functions have domain limits. Square root needs nonnegative input. Logarithm needs positive input. Reciprocal cannot use zero.
6. Can this calculator graph trigonometric functions?
Yes. It supports sine and cosine. You can choose radians or degrees for parent input values used in those functions.
7. What does the CSV download include?
The CSV file includes x values, internal inputs, parent outputs, and transformed outputs. It is useful for spreadsheets and records.
8. Is the PDF report printable?
Yes. The PDF includes the equation, key settings, notes, and the generated table. It can be saved or printed for study.